Find Inverse Of Square Root Functions

Examples, with detailed solutions, on how to find the inverse of of square root functions as well as their domain and range.

Example 1: Find the inverse function, its domain and range, of the function given by

f(x) = √(x - 1)

Solution to example 1:

  • Note that the given function is a square root function with domain [1 , + ∞) and range [0, +∞). We first write the given function as an equation as follows

    y = √(x - 1)

  • Square both sides of the above equation and simplify

    y 2 = (√(x - 1)) 2

    y 2 = x - 1

  • Solve for x

    x = y 2 + 1

  • Change x into y and y into x to obtain the inverse function.

    f -1(x) = y = x 2 + 1

    The domain and range of the inverse function are respectively the range and domain of the given function f. Hence

    domain and range of f -1 are given by: domain: [0,+ ∞) range: [1 , + ∞)


Example 2: Find the inverse, its domain and range, of the function given by

f(x) = √(x + 3) - 5

Solution to example 2:

  • Let us first find the domain and range of the given function.

    Domain of f: (x + 3) ≥ 0 which gives x ≥ - 3 and in interval form [- 3 , + ∞)

    Range of f: [- 5 , +∞)

  • Write f as an equation.

    y = √(x + 3) - 5

    which gives √(x + 3) = 5 + y

  • Square both sides of the above equation and simplify.

    (√(x + 3)) 2 = (5 + y) 2

    (x + 3) = (5 + y) 2

  • Solve for x.

    x = (5 + y) 2 - 3

  • Interchange x and y to obtain the inverse function

    f -1(x) = y = (5 + x) 2 - 3

    The domain and range of the inverse function are respectively the range and domain of the given function f. Hence

    domain and range of f -1 are given by: domain: [- 5,+ ∞) range: [- 3 , + ∞)


Example 3: Find the inverse, its domain and range, of the function given by

f(x) = - √(x 2 -1) ; x ≤ -1

Solution to example 3:

  • Function f given by the formula above is an even function and therefore not a one to one if the domain is the set R. However the domain in our case is given by x ≤ - 1 which makes the given function a one to one and therefore has inverse.

    Domain of f: (- ∞ , - 1] , given

    Range:
    For x in the domain (- ∞ , - 1] , the range of x 2 - 1 is given by [0,+∞), which gives a range of f(x) = - √(x 2 -1) in the interval (- ∞ , 0].

  • Write f as an equation, square both sides and solve for x, and find the inverse.

    y = - √(x 2 -1)

    y 2 = (- √(x 2 -1)) 2

    y 2 = x 2 -1

    x 2 = y 2 + 1

    x = ~+mn~√(y 2 + 1)

  • We now apply the domain of f given by x ≤ -1 to select one of the two solutions above. Hence

    x = - √(y 2 + 1)
  • Change x into y and y into x to obtain the inverse function.

    f-1(x) = y = - √(x 2 + 1)

    The domain and range of f -1 are respectively given by the range and domain of f found above

    Domain of f -1 is given by: [0 , + ∞) and its range is given by: (- ∞ , -1]




Exercises: Find the inverse, its domain and range, of the functions given below

1. f(x) = -2 √(x + 2) - 6

2. g(x) = 2 √(x 2 - 4) + 4 ; x ≥ 2


Answers to above exercises:

1. f -1(x) = (1/4)(x + 6) 2 - 2 ; domain: (-∞ , - 6] Range: [- 2 ; ∞)

2. g -1(x) = √((y - 4) 2 / 4 + 4) ; domain: [4 , +∞) Range: [2 , +∞)

More links and references related to the inverse functions.



Find the Inverse of a Square Root Function

Find the Inverse Functions - Calculator

Applications and Use of the Inverse Functions

Find the Inverse Function - Questions

Find the Inverse Function (1) - Tutorial.

Definition of the Inverse Function - Interactive Tutorial

Find Inverse Of Cube Root Functions.

Find Inverse Of Square Root Functions.

Find Inverse Of Logarithmic Functions.

Find Inverse Of Exponential Functions.






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